Generalized Invariance Principles for Stochastic Dynamical Systems and Their Applications

Investigating long-term behaviors of stochastic dynamical systems often requires to establish criteria that are able to describe delicate dynamics of the considered systems. In this article, we develop generalized invariance principles for continuous-time stochastic dynamical systems. Particularly, in a sense of probability one and by the developed semimartingale convergence theorem, we not only establish a local invariance principle, but also provide a generalized global invariance principle that allows the sign of the diffusion operator to be positive in some bounded region. We further provide an estimation for the time when a trajectory, initiating outside a particular bounded set, eventually enters it. Finally, we use several representative examples, including stochastic oscillating dynamics, to illustrate the practical usefulness of our analytical criteria in deciphering the stabilization or/and the synchronization dynamics of stochastic systems.